PDE Seminar

Spreading and Vanishing in Nonlinear Diffusion Problems with Free Boundaries

Du

Abstract

We consider nonlinear diffusion problems of the form \(u_t=u_{xx}+f(u)\)
with free boundaries. Such problems may be used to describe the
spreading of a biological or chemical species, with the free boundary
representing the expanding front. For any \(f(u)\) which is \(C^1\) and
satisfies \(f(0)=0\), we show that every bounded positive solution
converges to a stationary solution as \(t\to\infty\). For monostable,
bistable and combustion types of nonlinearities, we obtain a complete
description of the long-time dynamical behavior of the problem.
Moreover, by introducing a parameter \(\sigma\) in the initial data, we
reveal a threshold value \(\sigma^*\) such that spreading
(\(\lim_{t\to\infty}u= 1\)) happens when \(\sigma>\sigma^*\), vanishing
(\(\lim_{t\to\infty}u=0\)) happens when \(\sigma